Affine combinations of adaptive filters
Renato Candido, Magno T. M. Silva, Vítor H. Nascimento. Affine combinations of adaptive filters. In: 42nd Annual Asilomar Conference on Signals, Systems, and Computers, 2008, Pacific Grove, CA. Proceedings of Asilomar-2008, 2008, pp.236–240.
We extend the analysis presented in  for the affine combination of two least mean-square (LMS) filters to allow for colored inputs and nonstationary environments. Our theoretical model deals, in a unified way, with any combinations based on the following algorithms: LMS, normalized LMS (NLMS), and recursive-least squares (RLS). Through the analysis, we observe that the affine combination of two algorithms of the same family with close adaptation parameters (step-sizes or forgetting factors) provides a 3 dB gain in relation to its best component filter. We study this behavior in stationary and nonstationary environments. Good agreement between analytical and simulation results is always observed. Furthermore, a simple geometrical interpretation of the affine combination is investigated. A model for the transient and steady-state behavior of two possible algorithms for estimation of the mixing parameter is proposed. The model explains situations in which adaptive combination algorithms may achieve good performance.
 N. J. Bershad, J. C. M. Bermudez, and J.-Y. Tourneret, “An affine combination of two LMS adaptive filters – transient mean-square analysis”, IEEE Transactions on Signal Processing, vol. 56, pp. 1853–1864, May 2008.
Adaptive filters, Affine combination, Steady-state analysis, Transient analysis, LMS algorithm.